Optical device for measuring the position of an object

ABSTRACT

The invention relates to an optical device for measuring the position of an object along a first axis, the object being subjected to light radiations emitted by a light source. The optical device comprises: an imaging system comprising an objective for collecting light radiations diffused by the object, the imaging system having an optical axis extending parallel to the first axis; a transmission mask having at least a first aperture and a second aperture, the first aperture and second aperture being spaced from each other along a second axis, perpendicular to the first axis, the transmission mask being arranged so as to let a first part of the radiations and a second part of the radiations which are diffused by the object pass through the first aperture and the second aperture respectively, while blocking a part of the radiations emitted by the light source which is not diffused by the object; and a detector adapted for generating an image including a first spot and a second spot representative of the first part and second part of the radiations impacting the detector plane, wherein variation of the position of the object relative to the object plane of the imaging system along the first axis causes variation of a position of the first spot and of the second spot relative to each other along the second axis.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.15/381,320, filed on Dec. 16, 2016, which claims priority from EuropeanPatent Application Serial No. 15307065.1, filed Dec. 18, 2015, thedisclosures of which are incorporated herein by reference.

TECHNICAL FIELD OF THE INVENTION

The present invention relates to an optical device for measuring theposition of one or several objects. The invention finds particularapplication for measuring positions of quasi-punctual objects, such asmicrobeads, with a high precision.

BACKGROUND ART

In order to study interactions between DNA molecules and othercomponents, such as proteins, it is known to submit the DNA molecules tostretching forces and to measure the elastic properties (i.e. relativeextension versus force characteristics) of the molecules.

Document US 2003/0027187 discloses for instance an apparatus for testinga DNA molecule wherein the molecule is anchored at one end to ananchoring surface and at the other end to a paramagnetic bead. Theapparatus comprises magnets for applying a force to the bead so as tocontrol the stretching and torsion of the molecule. The apparatus alsocomprises a light source, a microscope and a camera for generating animage of the bead, as well as a computer for analyzing the imagegenerated.

Analysis of the image of the bead allows determining in real time theposition of the bead in three dimensions (x, y, z), and thus theextension of the molecule and the applied stretching force.

The x, y coordinates of the bead may be determined by using the symmetryof the bead and determining its center by auto-convolution. Indeed, thisfunction presents a maximum positive value which position is shifted by(2δx, 2δy) where δx(δy) is the shift of the bead image along x(y) fromthe original image center. Auto-convolution may be computed rapidlyusing a FFT algorithm and the maximum position may be obtained bylocally fitting a second order polynomial.

The z coordinate of the bead (i.e. coordinate of the bead along themagnification axis of the microscope) may be determined by comparing thediffraction pattern of the bead to a set of reference diffractionpatterns previously acquired during a calibration phase.

Indeed, interferences between light radiations emitted by the lightsource and light radiations diffused by the bead generate diffractionrings in the image recorded by the camera. The size of the diffractionrings varies with the distance of the bead relative to the focal planeof the microscope.

Calibration of the apparatus consists in recording several images of thebead by varying focusing of the microscope while keeping the bead in afixed position relative to the anchoring surface. This calibration phaseallows generation of different reference images of the beadcorresponding to different distances between the bead and the focusplane.

Once the calibration phase has been completed, comparison of the currentimage of the bead with the reference images, allows measurement of theposition of the bead with a precision of few nanometers. For instance,the method allows following positions of a few dozen of beads, with aprecision of about 3 nanometers between two video images. Inapplications wherein the apparatus is used for measuring the length ofthe DNA molecule, this allows localizing a sequence component of the DNAmolecule to within few nucleobases.

However, the calibration phase is time consuming, requires largecomputing resources and must be carried out for each bead separately.

Moreover, the proposed method requires the use of a large number ofpixels on the camera used to image one bead, especially if one wishes totest long DNA molecules because of the increasing of the size of theimage of the object. This may limit the number of beads one is able toanalyze (for instance 1000 beads for a 4 megapixels camera).

Therefore, the proposed method may not be extended for simultaneouslymeasuring positions of a large number of beads, such as thousands ofbeads for instance.

In addition, in order to vary focusing of the microscope, thecalibration phase requires the use of a high precision nano-positioningstage, including piezoelectric actuators, for moving the anchoringsurface relative to the microscope in a precise and repeatable way.

SUMMARY OF THE INVENTION

One aim of the invention is to provide an optical device for measuring aposition of an object with a high precision, which does not necessitatea calibration phase.

According to a first aspect, the invention provides an optical devicefor measuring the position of an object along a first axis, the objectbeing subjected to light radiations emitted by a light source, theoptical device comprising:

-   -   an imaging system comprising an objective for collecting light        radiations diffused by the object, the objective having an        optical axis extending parallel to the first axis,    -   a transmission mask having at least a first aperture and a        second aperture, the first aperture and second aperture being        spaced from each other along a second axis, perpendicular to the        first axis, the transmission mask being arranged so as to let a        first part of the radiations and a second part of the radiations        which are diffused by the object pass through the first aperture        and the second aperture respectively, while blocking a part of        the radiations emitted by the light source which is not diffused        by the object,    -   a separating arrangement for separating the first part of the        radiations from the second part of the radiations in opposite        directions along a third axis, perpendicular to the first and        second axes, and    -   a detector having a detector plane, the detector being adapted        for generating an image including a first spot and a second        spot, the first spot and the second spot being representative of        the separated first part and second part of the radiations        impacting the detector plane,

wherein variation of the position of the object relative to an objectplane of the imaging system along the first axis causes variation of aposition of the first spot and of the second spot relative to each otheralong the second axis.

In such an optical device, the transmission mask allows selecting tworadiation parts from the light radiations diffused by the object so asto generate two spots in the image recorded by the detector. As thedistance between the two spots along the second axis is proportional tothe distance between the bead and the object plane of the imagingsystem, it is possible to infer, from the image, the position of thebead according to the first axis.

In addition, the transmission mask blocks the part of the radiationscoming directly from the source. Therefore, radiations coming directlyfrom the source do not reach the objective so that only radiations whichhave been diffused (i.e. deviated) by the object may be observed in theimage generated by the detector. This greatly improves the contrast ofthe image and therefore increases the precision of the measurements.

The separating arrangement spatially shifts the two radiation partsapart from each other along the third axis. This prevents overlapping ofthe two spots when the two radiation parts cross each other in the planeof the detector.

Moreover, as the two spots are shifted along the third axis, it ispossible to determine whether the two radiation parts cross each otherbefore or after the detector plane so as to discriminate betweenpositive and negative values of the position of the image plane of thebead relative to the detector plane.

In addition, the optical device may have the following features:

-   -   according to a first embodiment, the transmission mask comprises        only two apertures,    -   the first aperture and the second aperture are arranged        symmetrically on opposite sides of the first axis,    -   the optical device comprises a processing module for processing        the image generated by the detector, the processing module being        configured for:

determining a position of a center of the first spot in the image,

determining a position of a center of the second spot in the image, and

computing the position of the object along the first axis as a functionof the position of the center of the first spot and of the position ofthe center of the second spot,

-   -   the processing module may be configured for determining the        position of the center of each spot by computing a maximum of        the auto-convolution of an average profile of the spot,    -   the processing module is configured for determining the position        of the object along the second axis and/or along the third axis        from positions of the spots on the image,    -   according to a second embodiment, the transmission mask        comprises a first pair of apertures dividing the first part of        the radiations into two first beams, and a second pair of        apertures dividing the second part of the radiations into two        second beams, and wherein the two first beams interfere with        each other so as to create a first interference pattern within        the first spot and the two second beams interfere with each        other so as to create a second interference pattern within the        second spot,    -   the optical device comprises a processing module for processing        the image generated by the detector, the processing module being        configured for determining a spatial phase shift between the        first interference pattern and the second interference pattern        along the second axis, and for determining a position of the        object along the first axis as a function of said spatial phase        shift,    -   determination of the spatial phase shift may comprise:

generating a first signature signal representative of a spatialvariation of the intensity of the first spot along the second axis,

generating a second signature signal representative of a spatialvariation of the intensity of the second spot along the second axis, and

determining a first reference point of the first signature signal wherea phase of the first signature signal is null near a maximum of anamplitude of the first signature signal determining a second referencepoint of the second signature signal where a phase of the secondsignature signal is null near a maximum of an amplitude of the secondsignature signal,

computing the spatial phase shift between the first interference patternand the second interference pattern as the distance along the secondaxis between the first point and the second point,

-   -   the processing module may also be configured for determining the        position of the object along the second axis from the first        reference point and the second reference point,    -   the processing module may also be configured for determining the        position of the object along the third axis, determination of        the position of the object along the third axis comprising:

generating a third signature signal representative of a spatialvariation of the intensity of the first spot and of the second spotalong the third axis,

computing an auto-convolution signal by auto-convolution of the thirdsignature signal,

determining a maximum of the auto-convolution signal, the coordinate ofthe maximum of the auto-convolution signal along the third axis y beingconsidered as being twice the position of the bead along the third axisy,

-   -   the optical device may comprise a first light source arranged to        emit light radiations toward the object according to a first        angle and a second light source arranged to emit light        radiations toward the object according to a second angle,    -   the first light source and the first pair of apertures may be        arranged such that a part of the light radiations emitted by the        first light source and diffused by the object passes through the        first pair of apertures while a part of the light radiations        emitted by the first light source but which is not diffused by        the object is blocked by the transmission mask,    -   the second light source and the second pair of apertures may be        arranged such that a part of the light radiations which is        emitted by the second light source and diffused by the object        passes through the second pair of apertures while a part of the        light radiations emitted by the second light source and which is        not diffused by the object is blocked by the transmission mask,    -   the light source may be a light source with a short length of        coherence (i.e. less than around 100 μm) such as a light        emitting diode (LED),    -   the separating arrangement may comprise at least one blade        having a face which is inclined relative to a plane        perpendicular to the first axis, so that the first part or the        second part of the light radiations which goes through the blade        is translated along the third axis,    -   the separating arrangement may comprise at least one prism        having a face which is inclined relative to a plane        perpendicular to the first axis, so that the first part or the        second part of the light radiations which goes through the prism        is deviated along the third axis,    -   the transmission mask may be located in a Fourier plane of the        imaging system or in a plane which is an image of the Fourier        plane of the imaging system through an optical relay, so as to        select parts of the light radiations which have been diffused by        the object according to predetermined angles.

According to a second aspect, the invention provides an optical devicefor measuring the position of an object along a first axis, the objectbeing subjected to light radiations emitted by a light source, theoptical device comprising:

-   -   an imaging system comprising an objective for collecting light        radiations diffused by the object, the objective having an        optical axis extending parallel to the first axis,    -   a transmission mask having at least a first pair of apertures        and a second pair of apertures, dividing the radiation diffused        by the object into two first beams passing through the first        pair of apertures and two second beams passing through the        second pair of apertures, while blocking a part of the        radiations emitted by the light source which is not diffused by        the object, and    -   a detector having a detector plane, the detector being adapted        for generating an image including a first spot and a second        spot, the first spot and the second spot being representative of        the first beams and second beams impacting the detector plane        respectively, the two first beams interfering with each other so        as to create a first interference pattern within the first spot        and the two second beams interfering with each other so as to        create a second interference pattern within the second spot.

wherein variation of the position of the object relative to an objectplane of the imaging system along the first axis causes spatial phaseshifting of first interference patterns and of the second interferencepattern relative to each other.

The transmission mask allows selecting two pairs of beams from the lightradiations diffused by the object so as to generate two spots havinginterference pattern in the image recorded by the detector. As thespatial phase shift of the interference patterns relative to each otherdepends on the distance between the bead and the object plane of theimaging system, it is possible to infer, from the image, the position ofthe bead according to the first axis with a very high precision. Theprecision of the measurement depends on the distance between twosuccessive interference fringes in the interferences patterns, and thusin part on the wavelength of the radiations emitted by the source.

In addition, the optical device may have the following features:

-   -   the optical device may comprise a processing module for        processing the image generated by the detector, the processing        module being configured for determining a spatial phase shift        between the first interference pattern and the second        interference pattern along the second axis, and for determining        a position of the object along the first axis as a function of        said spatial phase shift,    -   determination of the spatial phase shift may comprise:

generating a first signature signal representative of a spatialvariation of the intensity of the first spot along the second axis,

generating a second signature signal representative of a spatialvariation of the intensity of the second spot along the second axis, and

determining a first reference point of the first signature signal wherea phase of the first signature signal is null near a maximum of theamplitude of the signal,

determining a second reference point of the second signature signalwhere a phase of the second signature signal is null near a maximum ofthe amplitude of the signal

computing the spatial phase shift between the first interference patternand the second interference pattern as the distance along the secondaxis between the first point and the second point,

-   -   the processing module may be configured for determining the        position of the object along the second axis from the first        reference point and the second reference point,    -   the processing module may be configured for determining the        position of the object along the third axis, determination of        the position of the object along the third axis comprising:

generating a third signature signal representative of a spatialvariation of the intensity of the first spot and of the second spotalong the third axis,

computing an auto-convolution signal by auto-convolution of the thirdsignature signal,

determining a maximum of the auto-convolution signal, the coordinate ofthe maximum of the auto-convolution signal along the third axis y beingconsidered as being twice the position of the bead along the third axisy,

-   -   the optical device may comprise a first light source arranged to        emit light radiations toward the object according to a first        angle and a second light source arranged to emit light        radiations toward the object according to a second angle,    -   the first light source and the first pair of apertures may be        arranged such that a part of the light radiations emitted by the        first light source and diffused by the object passes through the        first pair of apertures while a part of the light radiations        emitted by the first light source but which is not diffused by        the object is blocked by the transmission mask,    -   the second light source and the second pair of apertures may be        arranged such that a part of the light radiations which is        emitted by the second light source and diffused by the object        passes through the second pair of apertures while a part of the        light radiations emitted by the second light source and which is        not diffused by the object is blocked by the transmission mask,    -   the light source may be a light source with a short length of        coherence (i.e. less than around 100 μm), such as a light        emitting diode (LED),    -   the separating arrangement may comprise at least one blade        having a face which is inclined relative to a plane        perpendicular to the first axis, so that the first part or the        second part of the light radiations which goes through the blade        is translated along the third axis,    -   the separating arrangement may comprise at least one prism        having a face which is inclined relative to a plane        perpendicular to the first axis, so that the first part or the        second part of the light radiations which goes through the prism        is deviated along the third axis,    -   the transmission mask may be located in a Fourier plane of the        objective or in a plane which is an image of the Fourier plane        of the objective through an optical relay, so as to select parts        of the light radiations which have been diffused by the object        according to predetermined angles.

The invention also relates to a method for measuring a position of anobject using a device as defined previously.

According to an embodiment of the invention, the object may be amagnetic bead.

According to an embodiment of the invention, a molecule having two endsis attached at one end to an anchoring surface and at the other end tothe magnetic bead, the device being positioned relative to the anchoringsurface so as to measure a distance between the magnetic bead and theanchoring surface.

According to an embodiment of the invention, a plurality of moleculesare attached, each molecule being attached at one end to the anchoringsurface and at the other end to an associated magnetic bead, and themethod comprises a step of generating an image showing a plurality ofpairs of spots, each pair of spots being generated by one of themagnetic beads, and scanning the image for determining for eachsuccessive pair of spots, a distance between the magnetic bead and theanchoring surface.

Moreover, the maximum size of an image of an object obtained with thisinvention is smaller than the maximum size of the image in the previousmethod described in the background art, especially for long travel ofthe object along the first axis. Thus, one aim of the invention is alsoto increase the number of objects whose position can be measured byusing a lower number of pixels for long travel range.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described with reference to the drawings, inwhich:

FIG. 1 diagrammatically shows a device used for measuring a position ofa bead attached to a DNA molecule,

FIG. 2 diagrammatically shows a device according to a first embodimentof the invention,

FIG. 3 diagrammatically illustrates two ray paths in the device of FIG.2,

FIG. 4 diagrammatically illustrates two spots generated in the detectorplane with the device of FIG. 2,

FIGS. 5 and 6 diagrammatically illustrates two different configurationsof the spots obtained for two positions of the bead,

FIG. 7 shows different images generated by the detector for differentpositions of the bead, obtained with the device of FIG. 2,

FIG. 8 shows an image generated by the detector for a batch of beads,obtained with the device of FIG. 2,

FIGS. 9A and 9B diagrammatically show a device according to a secondembodiment of the invention, comprising an amplitude mask, a separatingarrangement comprising two blades and, wherein the imaging systemcomprises an infinity-corrected objective and a lens tube,

FIGS. 10A, 10B and 10C diagrammatically show a device according to athird embodiment of the invention comprising an amplitude mask, aseparating arrangement comprising two blades, and wherein the imagingsystem comprises a finite objective,

FIGS. 11A and 11B diagrammatically show a device according to a fourthembodiment of the invention comprising an amplitude mask, a separatingarrangement comprising two prisms, and wherein the imaging systemcomprises an infinity-corrected objective and a tube lens,

FIG. 12 diagrammatically shows how the transmission mask blocksradiations coming directly from the light source,

FIG. 13 diagrammatically shows a device according to a fifth embodimentof the invention, comprising an amplitude mask and a separatingarrangement comprising two blades, and wherein the imaging systemcomprises an infinity-corrected objective, a 4 F arrangement and a lenstube,

FIG. 14 diagrammatically shows a device according to a sixth embodimentof the invention, comprising an amplitude mask, a separating arrangementcomprising two prisms, and wherein the imaging system comprises aninfinity-corrected objective, a 4 F arrangement and a tube lens,

FIG. 15 shows different images generated by the detector for differentpositions of the bead, obtained with the device of FIG. 13,

FIG. 16 shows an image generated by the detector for a batch of beads,obtained with the device of FIG. 15,

FIG. 17 is a diagram showing images generated by the detector whenvarying focusing of the optical objective,

FIGS. 18A and 18B are diagrams showing variation of the position of thefringes of the first spot and of the second respectively,

FIGS. 19A and 19B are diagrams showing different signals generated bythe processing module when processing the images of the spots: the firstsignal is a Fourier transform of the averaged interference profile ofeach spot, the second signal is obtained by band-pass filtering thefirst signal, and the third signal is the phase signal of the secondsignal,

FIGS. 20A and 20B are diagrams showing the position of the fringespattern measured using the phase signal (diamonds full lines) or themaximum of amplitude (circle dash lines) for the first spot and for thesecond spot respectively,

FIGS. 21A and 21B are diagrams showing respectively an error signal(that is the value of the signal minus the mean value of each step) anda corresponding distribution of the error signal,

FIG. 22 shows two signals allowing to determine a position of a beadalong the second axis x: the bottom signal corresponds to a mean profileof two fringes patterns, and the top signal is obtained byauto-convolution of the bottom signal,

FIGS. 23 and 24 are diagrams illustrating different steps carried out bya processing module for processing an image generated by an opticaldevice, according to the first embodiment of the invention,

FIGS. 25 to 27 are diagrams illustrating different steps carried out bya processing module for processing an image generated by an opticaldevice, according to the second embodiment of the invention.

DETAILED DESCRIPTION OF ONE EMBODIMENT OF THE INVENTION

In the example illustrated on FIG. 1, the optical device 1 according tothe invention is used for detecting the configuration of a molecule,such as a DNA molecule.

In this example, the device 1 comprises a light source 2, a magnetictweezer 3, a support 4, an optical imaging system 5, a detector 6 and aprocessing module 7.

The light source 2 comprises a monochromatic light-emitting diode (LED)adapted for emitting light radiations. Different wavelengths may be useddepending on the desired accuracy of the device. However, shortwavelength may damage the DNA molecule.

Therefore, the wavelength of the radiation is preferably comprisedbetween 200 nanometers (frequency 1505 THz) and 1 millimeter (frequency300 GHz).

The magnetic tweezer 3 comprises two magnets 31 and 32, which may bepermanent magnets.

The support 4 is made of a material which is transparent to the lightradiations emitted by the light source 2, such as glass for instance.The support 4 has an anchoring surface 41 to which the DNA molecule isattached. The support 4 may be a part of a capillary tube allowingintroduction of a liquid solution in which the DNA molecule is immersed.In that case, the anchoring surface 41 is an internal surface of thecapillary tube.

As illustrated on FIG. 1, the DNA molecule is attached at one end to amagnetic (paramagnetic or ferromagnetic) bead B and at the other end tothe anchoring surface 41. The magnetic bead B may be a polystyrene beadwith incorporated ferrite. The magnetic bead B may have a diameter offew micrometers or less. The DNA molecule may be labelled with biotinand digoxigenin at its ends, while the bead B is coated withstreptavidin and the anchoring surface 41 is coated with antidigoxigeninantibodies.

The optical imaging system 5 is adapted for collecting light radiationsdiffused by the bead B. The optical imaging system 5 has a magnificationaxis, which is also the optical axis O of the optical device 1. Theoptical axis O is parallel to a first direction z. The anchoring surface41 extends substantially perpendicularly to the magnification axis O ofthe optical objective 5.

The two magnets 31 and 32 are arranged at a distance from the anchoringsurface 41, on opposite sides of the magnification axis O. The magnets31, 32 are designed for generating a magnetic field so as to apply amagnetic force on the bead B and consequently a stretching force on theDNA molecule. By moving the magnets 31, 32 closer to or farther from theanchoring surface 41, it is possible to adjust the magnetic fieldgradient and thus control the stretching force applied to the molecule.

The detector 6 is interposed between the imaging system 5 and theprocessing module 7. The detector 6 may be a charge-coupled device (CCD)or a complementary metal-oxide-semiconductor (CMOS) camera. The detector6 is arranged to receive light radiations transmitted by the opticalimaging system 5 and to generate a corresponding image.

The processing module 7 may be a PC computer, a processor, an electroniccard, a dedicated integrated circuit or a programmable electroniccomponent. The processing module 7 is configured to analyze the imagegenerated by the detector 6 and determine a position of the bead B basedon the generated image.

FIG. 2 illustrates a first embodiment of the optical device 1 accordingto the invention. In this figure, the optical imaging system 5 comprisesan optical finite-objective 53 (i.e. an optical objective which isconfigured to provide an image of an object at a finite distance). OnFIG. 2, the optical finite-objective 53 has been represented in asimplified manner, as a single lens.

As illustrated on FIG. 2, the optical device 1 comprises a transmissionmask 8 and a separating arrangement 9.

The transmission mask 8 is located in the Fourier plane of the opticalobjective 53 of the imaging system 5.

The transmission mask 8 has a first aperture 81 and a second aperture82. In the example illustrated on FIG. 2, the first aperture 81 andsecond aperture 82 consist of a first slot and a second slot. The slots81 and 82 are located symmetrically on opposite sides of the opticalaxis O and are spaced from each other along a second axis x,perpendicular to the first axis z. In addition, both the first slot 81and second slot 82 have a length which extends parallel to a third axisy, perpendicular to the first axis z and the second axis x.

The transmission mask 8 is arranged so as to let a first part R1 of theradiations emitted by the source 21 and diffused by the bead B passthrough the first aperture 81 and a second part R2 of the radiationsemitted by the source 21 and diffused by the bead B pass through thesecond aperture 82. The transmission mask 8 is also arranged to block athird part R3 of the radiations which is directly emitted by the lightsource 2 (i.e. part of the radiations which has not been diffused by thebead). In other words, the transmission mask is arranged to select twoparts R1 and R2 of the radiations diffused by the bead B.

The separating arrangement 9 is arranged for spatially separating thefirst part R1 of the radiations and the second part R2 of the radiationswhich have been selected by the amplitude mask 8. The first part R1 andthe second part R2 of the radiations are shifted by the separatingarrangement 9 in opposite directions along the third axis y,perpendicular to the first and second axes z and x. As a result, thefirst part R1 of the radiations and the second part R2 of the radiationdo not meet with each other. This prevent the first part R1 and thesecond part R2 of the radiations from interfering with each other.

The separated first part R1 and second part R2 of the radiations impactthe plane 61 of the detector 6, generating respectively a first spot S1and a second spot S2. Due to the presence of the separating arrangement9, the first spot S1 and the second spot S2 do not overlap, even whenthe first part R1 and second part R2 of the radiations have the sameposition along the second axis x in the detector plane 61.

In FIG. 3, the first part R1 and second part R2 of the radiations havebeen represented in a simplified manner as two individual rays r₁ andr₂. The first spot S1 and the second spot S2 have also been representedin a simplified manner, as two impact points s₁ and s₂. As shown on thisfigure, the transmission mask 8 allows selecting the two rays r₁ and r₂according to their respective angles α relative to the optical axis O.Rays r₁ and r₂ impact the detector plane respectively at two impactpoints s₁ and s₂ separated from the optical axis of distances x′₁ andx′₂ along the second axis x.

The first point s₁ and the second point s₂ represent two images of thebead, whose coordinate in the detection plane along the second axis xare x₁′ and x₂′ respectively. The first point s₁ and the second point s₂are separated from each other by a distance x₁′-x₂′ along the secondaxis x. As shown on FIG. 3, positions of the impact points s₁ and s₂ arelinked to the distance dz between the bead and the object plane of theimaging system, as follows:

x ₁ =dz·tan α

x ₂ =−dz·tan α

where dz is the distance between the bead and the object plane of theimaging system along the first axis z and α is the opening angle for theselected rays r₁ and r₂.

Thus, since the first point s₁ and the second point s₂ are in the imageplane of the imaging system, their positions x′₁ and x′₂ in the imageplane are given by the following relationships:

x ₁ ′=g _(y) ·x ₁

x′ ₂ =g _(y) ·x ₂

where g_(y) is the lateral magnification of the optical imaging system,and α is the angle of the selected rays relative to the optical axis O.

The distance x₁′-x₂′ between the two impact points s₁ and s₂ of the tworays r₁ and r₂, in the detector plane is proportional to the position ofthe bead along the first axis z, and is given by:

${dz} = {\frac{1}{2\; g_{y}\tan \; \alpha}\left( {x_{2}^{\prime} - x_{1}^{\prime}} \right)}$

where g_(y) is the lateral magnification of the imaging system, and α isthe angle of the selected rays relative to the optical axis O.

The sensitivity of the optical device 1 is therefore proportional to themagnification and to the tangent of the aperture angle of the opticalimaging system, while the numerical aperture defined by n sin α definesthe maximum possible angles.

At the same time, the position of the bead along the second axis x canbe deduced with the following relationship:

$x = \frac{x_{1}^{\prime} + x_{2}^{\prime}}{2\; g_{y}}$

where x is the coordinate of the bead along the second axis x.

If the case that the slots 81 and 82 are not properly centered, leadingto different angles, and for a bead which would not be placed on theoptical axis O, then the relations would become:

x ₁ −x=dz·tan α₁

x ₂ −x=−dz·tan α₂

where tan α₁ and tan α₂ are the slightly different selected angles, thenthe measured signal is:

${dz} = {\frac{x_{2}^{\prime} - x_{1}^{\prime}}{g_{y}} \cdot \frac{1}{{\tan \; \alpha_{1}} + {\tan \; \alpha_{2}}}}$

This relationship ensures that the measurement of dz remains uncoupledwith the measurement of x even in the case of a slight uncertainty inthe mounting of the slots.

$x = {\frac{x_{1}^{\prime} + x_{2}^{\prime}}{2\; g_{y}} + {{dz}\frac{{\tan \; \alpha_{2}} - {\tan \; \alpha_{1}}}{2}}}$

If tan α₁ and tan α₂ are not perfectly known, the x coordinate might beslightly polluted by the z variations. Tuning α₁ and α₂ so that theiramplitude is really equal reduces this defect.

As illustrated on FIG. 4, when the bead B is located in the object planeof the imaging system, the image of the bead is located at the imageplane of the imaging system, which is also the detector plane, so thatx₁=x₂=0. However, the first image S1 of the bead is translated by theseparating arrangement 9 along the third axis y by a constant value +y₀while the second image S2 of the bead is translated by the separatingarrangement 9 along the third axis y by a constant value −y₀. Thisallows to avoid overlapping of the spots S1 and S2 and preciselymeasuring the position z of the bead around the object plane of theimaging system.

FIGS. 5 and 6 illustrate two images generated by the detector 6 for twodifferent values of the position of the bead along the first axis z.Each image includes two spots S1 and S2. In both images, the spots areseparated from each other by a constant distance equal to 2y₀ along thethird axis y.

The processing module 7 (illustrated on FIG. 1) is configured forprocessing the image generated by the detector 6 so as to determine aposition of the bead B along axes x, y and z from the image generated bythe detector 6.

To this end, the processing module 7 carries out the steps illustratedon FIG. 23:

-   -   according to a first step 201, the processing module 7        determines a position of a center of the first spot S1 in the        image (i.e. coordinates x′₁ and y′₁),    -   according to a second step 202, the processing module 7        determines a position of a center of the second spot S2 in the        image (i.e. coordinates x′₂ and y′₂), and    -   according to a third step 203, the processing module 7 computes        the position of the bead B as a function of the positions of the        center of the first spot and of the position of the center of        the second spot.

The position of the bead along the third axis y is computed as:

$y = \frac{y_{1}^{\prime} + y_{2}^{\prime}}{2\; g_{y}}$

where (x₁′, y₁′) are coordinates of the center of the first spot S1,(x₂′,y₂′) are coordinates of the center of the second spot S2 in theplane of the detector and α is the angle of the selected radiationsrelative to the optical axis O.

As illustrated on FIG. 24, the position of the center of each spot isdetermined by computing the autoconvolution of one-dimensional profilesextracted from the image. For each spot and for the two axis x and y, aone-dimensional profile is extracted from the image (step 301) byaveraging the pixels of the image along the perpendicular axis (forexample, to compute the position x′₁, the lines of pixels of a part ofthe image which contains the spot S1 are averaged which provides aone-dimensional signal along x for the spot S1). Then for each extractedone-dimensional profile, an autoconvolution signal is computed (step302) by computing the inverse Fourier Transform of the square of theFourier Transform of the one dimensional profile.

The auto-convolution signal resulting from such an auto-convolutionpresents a positive maximum which position offset versus the center ofthe extracted profile is twice the offset of the spot from the center ofthe profile. Thus by determining this offset by locally fitting theauto-convolution signal with a second order polynomial for the fourone-dimensional profiles (step 303), the processing module 7 determinesthe four coordinates x′₁, y′₁, x′₂ and y′₂ (steps 304 and 305).

The precision of the measurement depends on the characteristic size ofthe spots and on the number of photons used to generate the spots. Thenumber of photons depends on the number of pixels covered by the spotand on the maximum possible number of detectable photons per pixelbefore saturation (i.e. “the well depth”).

Precision of the determination of the center of a spot is proportionalto:

$\frac{\sigma}{\sqrt{N}}$

where σ is the width at half maximum of the spot size, the shape of thespot being considered as Gaussian in a first approximation, and N is thetotal number of photons contained in the spot. σ is determined by thedimensions of the slots letting the radiations pass through thetransmission mask.

FIG. 7 shows different images which have been generated by the detector6 for different positions of the beads along the axis z. The bead was aparamagnetic bead having a diameter of 1 micrometer. The images havebeen obtained with an optical imaging system having a magnificationfactor g_(y) of 40, a transmission mask having two slots and aseparating arrangement comprising two inclined glass blades arrangedsymmetrically around the axis x.

FIG. 8 shows an image which has been generated by the detector for abatch of beads. Each bead was a paramagnetic bead having a diameter of 1micrometer. The image has been obtained with an optical objective havinga magnification factor g_(y) of 20, a transmission mask having two slotsand a separating arrangement comprising two inclined glass bladesarranged symmetrically around the axis x. The beads were placed in amicrofluidic chamber substantially at the same position according to theaxis z. The field of view was 628×628 micrometers. The image showsseveral pairs of spots, each pair of spots corresponding to one bead.

FIGS. 9A and 9B illustrate a second embodiment of an optical device 1according to the invention.

In this second embodiment, the transmission mask 8 comprises two pairsof apertures 81, 83 and 82, 84.

The transmission mask 8 comprises a first pair of apertures 81, 83dividing the first part R1 of the radiations into two first beams R11and R12, and a second pair of apertures 82, 84 dividing the second partR2 of the radiations into two second beams R21 and R22. The first pairof apertures 81, 83 consists of a first pair of slots and the secondpair of apertures 82, 84 consists of a second pair of slots.

In addition, the device comprises two light sources 21 and 22 arrangedfor illuminating the bead B according to two different angles. The lightsources 21 and 22 are arranged so as to direct light radiations towardthe bead B according to a first angle and according to a second anglerespectively.

More precisely, the first light source 21 generates first lightradiations which are directed between the apertures 81, 83 of the firstpair of apertures. In the same way, the second light source 22 generatessecond light radiations which are directed between the apertures 82, 84of the second pair of apertures.

With this arrangement, a part of the first light radiations which isdiffused by the bead B is allowed to pass through the apertures 81, 83of the first pair of apertures, while the part of the first lightradiations coming directly from the first light source 21 impacts theamplitude mask 8 between the apertures 81, 83 and is blocked.

Similarly, the second light source 22 is arranged so that a part of thesecond light radiations which is diffused by the bead B is allowed topass through the apertures 82, 84 of the second pair of apertures, whilethe part of the second light radiations coming directly from the secondlight source 22 impacts the amplitude mask between the apertures 82, 84and is blocked.

In the second embodiment shown on FIGS. 9A and 9B, the optical imagingsystem 5 comprises an infinity-corrected objective 53 (i.e. the opticalobjective has an image distance that is set to infinity) and a tube lens54. In other words, the image of an object positioned at the objectfocal plane of the optical objective 53 (which is also the object planeof the optical imaging system 5) does not form an image but is convertedby the optical objective 53 into an infinity parallel beam. The tubelens 54 is adapted for focusing parallel beams produced by the opticalobjective 53 on the image plane of the optical imaging system 5.

As illustrated on FIGS. 9A and 9B, the separating arrangement 9comprises two blades 91, 92.

Each blade has parallel faces which are inclined relative to a plane x,y perpendicular to the first axis z. More precisely, the blades includea first blade 91 which is inclined of a first angle and a second blade92 which is inclined of a second angle, opposite to the first angle. Thefirst blade 91 and the second blade 92 are arranged so that the firstpart of the light radiations goes through the first blade 91 and thesecond part of the radiation goes through the second blade 92. In thismanner, the first beams R11, R12 and the second beams R21, R22 comingfrom the apertures 81, 83 and 82, 84 respectively are translated alongthe third axis y in opposite directions.

The tube lens 54 is adapted for focusing each infinity parallel beamR11, R12, R21, R22 on an image plane.

The two first beams R11 and R12 impact the detector plane, thus forminga first spot S1. The two second beams R21 and R22 impact the detectorplane, thus forming a second spot S2.

Moreover, the two first beams R11 and R12 interfere with each other soas to create a first interference pattern within the first spot S1.Similarly, the two second beams R21 and R22 interfere with each other soas to create a second interference pattern within the second spot S2.

FIGS. 10A, 10B and 10C diagrammatically show a device according to athird embodiment of the invention.

According to this third embodiment, the optical imaging system 5 onlycomprises a finite objective 53 (i.e. an optical objective which has animage distance that is finite). In other words, the image of an objectpositioned at the object plane of the optical objective form an imagepositioned in an image plane.

As in the first embodiment, the separating arrangement 9 comprises twoblades 91 and 92.

As in the first embodiment, each blade has parallel faces which areinclined relative to a plane x, y perpendicular to the first axis z.More precisely, the blades include a first blade 91 which is inclined ofa first angle and a second blade 92 which is inclined of a second angle,opposite to the first angle. The first blade 91 and the second blade 92are arranged so that the first part of the light radiations goes throughthe first blade 91 and the second part of the radiation goes through thesecond blade 92. In this manner, the first beams R11, R12 and the secondbeams R21, R22 coming from the apertures 81, 83 and 82, 84 respectivelyare translated along the third axis y in opposite directions.

As shown on FIGS. 10A and 10B, the optical imaging system 5 has a backfocal plane 51 (or Fourier plane) and an object plane 52. Thetransmission mask is positioned in the back focal plane 51 of theimaging system 5 so as to select parts of the light radiations whichhave been diffused by the object according to predetermined angles.Depending on the numerical aperture of the optical objective, thepredetermined angles may be equal or greater than 45° or 50°.

As shown on FIG. 10C, the maximum distance d1 between the blades 91, 92and the back focal plane 51 along the optical axis O is such that allfirst beams R11 and R12 go through the first blade 91 and all secondbeams R21 and R22 go through the second blade 92.

The maximum distance d1 depends on the distance d2 between the aperture83 (or 82) which is the closest to the optical axis O and the opticalaxis O.

The plane where the separating arrangement is located is determined bythe maximum angle θ of the radiations emitted by the bead when the beadis located in the border of the field-of-view of the optical objectiveat this plane, and by the position p along the second axis x of the partof the transmission mask which is closest to the first axis, the maximumdistance being

${d\; 1} = {\frac{p}{\tan \; \theta}.}$

FIGS. 11A and 11B diagrammatically show a device according to a fourthembodiment of the invention wherein the imaging system 5 comprises aninfinity-corrected objective 53 and a tube lens 54.

In this embodiment, the separating arrangement 9 comprises two prisms93, 94.

The first prism 93 and the second prism 94 are positioned between thetransmission mask 8 and the tube lens 53. The prisms 93 and 94 arearranged so that the first part R1 of the light radiations goes throughthe first prism 93 and is deviated by the first prim 93. The second partR2 of the radiation goes through the second prism 94 and is deviated bythe second prism 94. Deviation of the two parts R1 and R2 of theradiations are converted into translations by the tube lens 53. In thismanner, the first beams R11, R12 and the second beams R21, R22 aretranslated along the third axis y in opposite directions.

As shown on FIG. 11B, the maximum distance d1 between the prisms 93, 94and the back focal plane 51 along the optical axis O is such that allfirst beams R11 and R12 go through the first prism 95 and all secondbeams R21 and R22 go through the second prism 94.

The maximum distance d1 depends on the distance d2 between the aperture83 (or 82) which is the closest to the optical axis O and the opticalaxis O.

FIG. 12 diagrammatically shows how the transmission mask 5 blocks lightradiations R3 coming directly from the light source, assuming the lightsource is positioned on the optical axis O.

Light radiations R3 which are emitted by the source and which are notdiffused by the bead B, propagate parallel to the optical axis. Theseradiations R3 are focused by the optical imaging system 5 in the backfocal plane 51, between the apertures 83 and 82, in an area of thetransmission mask which is opaque to the radiations.

FIG. 13 diagrammatically shows a device according to a fifth embodimentof the invention.

According to this fifth embodiment, the device 1 comprises an imagingsystem 5. The optical imaging system 5 comprises an infinity-correctedobjective 53, an optical relay 10 and a tube lens 54.

In this fifth embodiment, the back focal plane 51 is located inside theoptical objective 53 and is not accessible for positioning the amplitudemask 8.

The optical relay 10 can be a 4 F arrangement. The 4 F arrangement 10 ispositioned between the optical objective 53 and the transmission mask 8.The 4 F arrangement 10 is configured to produce an image plane 51′ ofthe back focal plane 51, outside the optical objective 53. The amplitudemask 8 is positioned in the image plane 51′.

The 4 F arrangement 10 comprises a first lens 101 having a first focallength 103 and a second lens 102 having a second focal length 104. Thefirst lens 101 and the second lens 102 may have different focal length.The first focal lens 102 is positioned relative to the optical objective53, so that the object focal plane of the first lens 101 is located inthe back focal plane 51 of the optical objective 53. The second lens 102is positioned relative to the first lens 101 so that the object focalplane of the second lens 102 is located in the back focal plane of thefirst lens 101. As a result, the image plane 51′ of the back focal plane51 of the optical objective 53 is located in the back focal plane of thesecond lens 102.

As in the second embodiment of FIGS. 9A and 9B, the device 1 alsocomprises a separating arrangement 9. The separating arrangementcomprises two blades 91 and 92, and the tube lend 54 is positionedbetween the amplitude mask 8 and the separating arrangement 9.

FIG. 14 diagrammatically shows a device 1 according to a sixthembodiment of the invention.

As in the fifth embodiment of FIG. 13, the device 1 comprises an imagingsystem 5 comprising an infinity-corrected objective 53, an optical relay10 and a tube lens 54.

However, in this sixth embodiment, the device 1 comprises a separatingarrangement comprising two prisms 93, 94, the prisms 93, 94 beingpositioned between the amplitude mask 8 and the tube lens 54.

FIG. 15 shows different images generated by the detector 6 for differentpositions of the bead B along the axis z, with the device of FIG. 13.

It is to be noted that similar images would be obtained with the devicesof FIGS. 9 to 12 and 14.

As shown in FIG. 15, the two first beams R11 and R12 interfere with eachother in the detector plane so as to create a first interference patternwithin the first spot S1. The two second beams R21 and R22 interferewith each other in the detector plane so as to create a secondinterference pattern within the second spot S2. Each interferencepattern comprises interferences fringes extending parallel to the axisy. Successive fringes of each interference pattern are spaced by aconstant pitch i.

The processing module 7 (illustrated on FIG. 1) is configured forprocessing the image generated by the detector 6 so as to determine aposition of the bead B in three dimensions, along axes x, y and z.

To this end, the processing module 7 determines a spatial phase shiftbetween the first interference pattern and the second interferencepattern along the second axis (x).

More precisely, the processing module carries out the steps illustratedon FIG. 23:

According to a first step 401, the processing module generates a firstsignature signal representative of a spatial variation of the intensityof the first spot (S1) along the second axis (x).

In the same way, the processing module generates a second signaturesignal representative of a spatial variation of the intensity of thesecond spot (S2) along the second axis (x).

Examples of a first signature signal and of a second signature signalsare shown on FIGS. 19A and 19B (signals illustrated at the top of thefigures).

According to a second step 402, the processing module filters the firstsignature signal and the second signature signal by applying a band-passfilter centered on the spatial frequency of the interference pattern.The spatial frequency is defined by the geometrical parameters of theoptical device (in particular, the distance between the slots of a pairof slots of the transmission mask).

Examples of a filtered first signature signal and of a filtered secondsignature signal are shown on FIGS. 19A and 19B (signals illustrated atthe middle of the figures).

According to a third step 403, the processing module applies a Hilberttransform to each filtered signal so as to generate a correspondingphase signal representative of the phase of the filtered signal. Thisstep allows generating a first phase signal representative of the phaseof the filtered first signal and of the phase of the filtered secondphase signal.

Examples of a first phase signal and of a second phase signal are shownon FIGS. 19A and 19B (signals illustrated at the bottom of the figures).

According to a fourth step 404, the processing module determines for thefirst phase signal, a position x₁ of the corresponding first spot alongthe x axis. The position of the first spot is along the axis x is theposition of a first reference point where the first phase signal is zeronear the maximum amplitude of the filtered first signature signal.

In the same way, the processing module determines for the second phasesignal, a position x₂ of the corresponding second spot along the x axis.The position of the second spot is along the axis x is the position of asecond reference point where the second phase signal is zero near themaximum amplitude of the filtered second signature signal.

The first reference point and the second reference point are identifiedby arrows on FIGS. 19A and 19B.

According to a fifth step 405, the processing module compute the spatialphase shift between the first interference pattern and the secondinterference pattern as the distance x₂−x₁ along the second axis (x)between the first reference point and the second reference point.

The position of the bead along the first axis z is computed as a linearcombination of x₁ and x₂. In particular, the position of the bead alongthe first axis z may be computed as proportional to the spatial phaseshift x₂−x₁.

The position of the bead along the second axis x is computed as a linearcombination of x₁ and x₂. In particular, the position of the bead alongthe second axis x may be computed as the mean of x₁ and x₂, i.e.x=½(x₁+x₂).

Alternatively, in order to determine the position of the bead along thefirst axis z and along the second axis x, the processing module 7 maycarry out the steps illustrated on FIG. 26.

According to a first step 501, the processing module 7 generates a firstsignature signal representative of a spatial variation of the intensityof the first spot (S1) along the second axis (x).

In the same way, the processing module generates a second signaturesignal representative of a spatial variation of the intensity of thesecond spot (S2) along the second axis (x).

According to a second step 502, the processing module 7 computes a firstFourier transform signal by applying a Fourier transform to the firstsignature signal.

In the same way, the processing module computes a second Fouriertransform signal by applying a Fourier transform to the second signaturesignal.

According to a third step 503, the processing module 7 determines aphase of the first signature signal from the first Fourier transformsignal. To this end, the phase of the first signature signal isdetermined by identifying an amplitude peak of the first Fouriertransform signal (corresponding to the spatial frequency of theinterference pattern of the first spot) and linear fitting the phase ofthe first Fourier transform signal around the peak. The constant term ofthe resulting regression line (intercept of the regression line) isconsidered as the phase of the first signature signal.

The constant term of the resulting regression line is considered to bethe phase of the first signature signal, but is undetermined by amultiple of 2π (for instance, the value of the phase signature signalmay be 0.1+2nπ while the correct value is 0.1). The linear term of theresulting regression line (slope of the regression line) is used todetermine a correction of the phase of the first signature signal.

In the same way, the processing module determines a phase of the secondsignature signal from the second Fourier transform signal. To this end,the phase of the second signature signal is determined by identifying anamplitude peak of the second Fourier transform signal (corresponding tothe spatial frequency of the interference pattern of the second spot)and linear fitting the phase of the second Fourier transform signalaround the peak. The constant term of the resulting regression line(intercept of the regression line) is considered as the phase of thesecond signature signal.

The constant term of the resulting regression line is considered to bethe phase of the second signature signal, but is undetermined by amultiple of 2π (for instance, the value of the phase signature signalmay be 0.1+2nπ while the correct value is 0.1). The linear term of theresulting regression line (slope of the regression line) is used todetermine a correction of the phase of the second signature signal.

In other words, the processing module uses the linear term of the linearequation of the regression line to find the multiple of 2π to be addedto correct the phase of the corresponding signature signal.

According to a fourth step 504, the processing module determines fromthe phase of the first signature signal, a position x₁ of thecorresponding first spot along the x axis.

In the same way, the processing module determines from the phase of thesecond signature signal, a position x₂ of the corresponding second spotalong the x axis.

According to a fifth step 505, the processing module computes thespatial phase shift between the first interference pattern and thesecond interference pattern as the distance x₂−x₁ along the second axis(x).

The position of the bead along the first axis z is computed as a linearcombination of x₁ and x₂. In particular, the position of the bead alongthe first axis z may be computed as proportional to the spatial phaseshift x₂−x₁.

The position of the bead along the second axis x is computed as a linearcombination of x₁ and x₂. In particular, the position of the bead alongthe second axis x may be computed as the mean of x₁ and x₂, i.e.x=½(x₁+x₂).

The position of the bead along the third axis y is determined asillustrated on FIG. 27.

According to a first step 601, the processing module 7 generates a thirdsignature signal representative of a spatial variation of the intensityof the first spot (S1) and of the second spot (S2) along the third axis(y).

According to a second step 602, the processing module 7 computes theauto-convolution of the third signature signal. The auto-convolution ofthe third signature signal is computed by inverse Fourier transformingthe square of the Fourier transform signal of the third signaturesignal.

According to a third step 603, the processing module 7 determines themaximum of the auto-convolution signal. Maximum of the auto-convolutionsignal is determined by locally fitting the auto-convolution signal by asecond-order polynomial function, and then determining a maximum of thesecond-order polynomial function.

The coordinate of the maximum of the auto-convolution signal isconsidered as being twice the position of the bead along the third axisy.

As the interference pattern is periodic, the characteristic length ofthe interference pattern is not the half-length of the spot σ (as in thefirst embodiment), but a fraction of the pitch of the fringe pattern(typically ¼ of the pitch, which is substantially inferior to σ).

In addition, computation of positions x′₁ and x′₂ of the centers of thespots along the axis x can be made by unidimensional averaging andunidimensional Fourier processing, which requires less computingresources than computing the radial bead profile in 2D involving nx·nysquare-root to be taken for all the pixels in the area nx ny around thebead.

Moreover, spreading of the spots along the axis x causes an increase ofthe number of pixels impacted by the light radiations, and thereforegives the possibility to dramatically increase the total number N ofphotons contained in each spot before saturation of the detector. As aresult, the precision of the measurement is increased.

Each spot contains fringes which may be viewed as a periodic collectionof spots, having a full width at half maximum of:

$\sigma = \frac{i}{4}$ $i = \frac{\lambda}{2\; \sin \; \beta}$

where i is the pitch between two successive the fringes, λ is thewavelength emitted by the source and β is the angle between two beamscoming from a pair of slots.

The precision of the computation of the center of a spot is thus:

$\frac{\lambda}{8\sqrt{N}\sin \; \beta}$

where N is the total number of useful photons contained in the spot,each fringe contributing to the precision of the measurement with thenumber of photon contained therein.

The number N of useful photons is the total number of photons containedin the image which contribute to the interferometric pattern. The numberN is:

N=N ₀ ·c·n _(x) ·n _(y)

where N₀ is the number of photons corresponding to the saturationthreshold of one pixel of the detector (i.e. the well depth), c is thecontrast of the fringes (quantifying the proportion of useful photons)and is comprised between 0 and 1, n_(x) and n_(y) are the number ofpixels according to the axes x and y defining the extent of the spot.

Finally, the precision of the computed z coordinate of the bead from twospots is:

$\frac{\lambda}{16\sqrt{N}\sin \; \beta \; \tan \; \alpha} = \frac{i}{8\sqrt{N}\tan \; \alpha}$

This precision can be adjusted by controlling simple parameters, such asthe object aperture angle (tan α), the spatial frequency of the fringes(sin β) and the area of the spot which sets, for a given detector, thetotal number N of photons contributing to the measurement of thecoordinate z of the bead.

FIG. 16 shows an image generated by the detector 6 for a batch of beads.

The image has been obtained with a red LED centered around 660nanometers and having a spectral width of 15 nanometers, a microscopeobjective having a magnification of 40, a 4 F arrangement having a firstlens having a focal length of 125 millimeters and a second lens having afocal length of 100 millimeters, a transmission mask having four slotsand a separating arrangement comprising a tube lens having a focallength of 100 millimeters and two inclined glass blades arrangedsymmetrically around the axis x. Each bead was a paramagnetic beadhaving a diameter of 1 micrometer. The distance between the two pairs ofslots was 2.5 millimeters, and the distance between two slots of a samepair of slots was 1.4 millimeters. Each slot had a width of 0.2millimeters (along the axis x) and a length of 1.3 millimeters (alongthe axis y). The blades were inclined of +10° and −10° relative to thex, y plane. The thickness of each blade was 1 millimeter. The beads wereplaced in a microfluidic chamber substantially at the same positionaccording to the axis z. The field of view was 400×400 micrometers. Theimage shows several pairs of spots, each pair of spots corresponding toone bead. Each spot contains an interference pattern made of parallelfringes.

The image can be scanned for determining for each successive pair ofspots, coordinates of the corresponding magnetic bead.

The proposed solution does not necessitate a calibration phase for eachbead. This greatly simplifies the measurements and eliminates the needfor a high precision nano-positioning stage. The device and the methodmay be implemented with a standard microscope objective.

FIG. 17 shows three images generated by the detector when varyingfocusing of the optical objective.

The bead was stuck to the surface anchoring, while the optical objectivefocus position was moved as shown on FIG. 17. As can been seen on thisfigure, the focus position remained constant for 10 s, it was thendecreased by 3 microns in steps of −0.1 microns for 30 s, the focusposition was then increased to +3 microns in 60 steps lasting 60 s, andthe focus was brought back to its position in 30 s.

FIGS. 18A and 18B are diagrams showing variation of the position of thefringes of the first spot and of the second respectively.

The image intensity of each fringes system is shown versus time. Bothare shifted along the axis x when the focus position moves but theirdisplacement are opposite in direction. Each line of the diagramscorresponds to the averaging of one of the fringe system over typically10 camera lines. These 1D profiles correspond to the signal that will beused at each frame to measure the bead position along the axes x and z.

FIGS. 19A and 19B are diagrams showing different signals generated bythe processing module when processing the images of the first spot (FIG.19A) and of the second spot (FIG. 19B).

For each spot, the first signal (at the top) is a Fourier transform ofthe averaged interference profile of the spot.

The second signal (in the middle) is obtained by band-pass filtering thefirst signal.

The third signal (at the bottom) is the phase signal of the secondsignal.

As can be seen in the insert of FIG. 19A, the Fourier spectrum of theseprofiles display a well defined peak (corresponding here to a modenumber of 13). The band-pass filter is centered on this maximum and itswidth is adjusted to allow just the peak to be keep only. The result ofthis filter is shown in full line together with the second signal. Byusing the Hilbert transform, the imaginary part (dashed lines)associated also to the band-filtered signal is obtained. Having acomplex signal is very convenient to measure the amplitude of the signal(in full line) and particularly to measure its phase. The filteredprofiles have a clear maximum that moves along x with the change offocus position in z. This amplitude maximum is marked by the verticalarrow. The third signal shows the phase profile. The phase signal varieslinearly with x, its slope being related to the spatial frequency of thefringes. The position where the phase is equal to zero for the signal inthe place where the amplitude is maximum defines x(φ=0) whichcharacterizes the fringes position, is marked by a vertical arrow. Thephase is multiply defines (modulo 2π) but the exact phase may berecovered either by continuity or by using the position of the amplitudemaximum as a coarse value.

FIGS. 20A and 20B are diagrams showing the position of the fringespattern measured using the phase signal (diamonds full lines) or themaximum of amplitude (circle dash lines) for the first spot and for thesecond spot respectively.

It is to be noted that the amplitude signal is roughly linear butpresents some irregularities while the phase signal is extremely linearwith minimal noise. Owing to small dissymmetry in the position of theslots in the amplitude mask, the slope of the two phase lines areslightly different.

FIG. 21A is a diagram showing an error signal: that is the value of thesignal minus the mean value of each step (the signal has not beenreported when the focus position was moving). On FIG. 21B, this errorsignal in x has been converted in a z signal. The z signal has aGaussian distribution with a σ of 1 nm.

FIG. 22 shows two signals. The bottom signal corresponds to the meanprofile over 64 points of two fringes pattern, each peak correspondingto a fringe pattern. These two peaks are shifted on the right by alittle more than 4 pixels. The top signal is the auto-convolution of thebottom signal, it presents three maxima with the strongest shifted bymore than 8 pixels. Finding the position of the auto-convolution maximumprovides twice the shift of the bottom signal. This allows tracking theposition of the bead according to the second axis x (which isperpendicular to the direction of the fringes).

1. An optical device for measuring the position of an object along afirst axis, the object being subjected to light radiations emitted by alight source, the optical device comprising: an imaging systemcomprising an objective for collecting light radiations diffused by theobject, the objective having an optical axis extending parallel to thefirst axis, a transmission mask having at least a first pair ofapertures and a second pair of apertures, the first pair of aperturesand second pair of apertures being spaced from each other along a secondaxis, perpendicular to the first axis, the transmission mask dividingthe radiation diffused by the object into two first beams passingthrough the first pair of apertures and two second beams passing throughthe second pair of apertures, while blocking a part of the radiationsemitted by the light source which is not diffused by the object, and adetector having a detector plane, the detector being adapted forgenerating an image including a first spot and a second spot, the firstspot and the second spot being representative of the first beams andsecond beams impacting the detector plane respectively, the two firstbeams interfering with each other so as to create a first interferencepattern within the first spot and the two second beams interfering witheach other so as to create a second interference pattern within thesecond spot, wherein variation of the position of the object relative toan object plane of the imaging system along the first axis causesspatial phase shifting of first interference patterns and of the secondinterference pattern relative to each other.
 2. The optical deviceaccording to claim 1, comprising a processing module for processing theimage generated by the detector, the processing module being configuredfor determining a spatial phase shift between the first interferencepattern and the second interference pattern along the second axis, andfor determining a position of the object along the first axis as afunction of said spatial phase shift.
 3. The optical device according toclaim 2, wherein determination of the spatial phase shift comprises:generating a first signature signal representative of a spatialvariation of the intensity of the first spot along the second axis,generating a second signature signal representative of a spatialvariation of the intensity of the second spot along the second axis, anddetermining a first reference point of the first signature signal wherea phase of the first signature signal is null near a maximum of theamplitude of the signal, determining a second reference point of thesecond signature signal where a phase of the second signature signal isnull near a maximum of the amplitude of the signal, computing thespatial phase shift between the first interference pattern and thesecond interference pattern as the distance along the second axisbetween the first point and the second point.
 4. The optical deviceaccording to claim 3, wherein the processing module is configured fordetermining the position of the object along the second axis from thefirst reference point and the second reference point.
 5. The opticaldevice according to claim 1, wherein the processing module is configuredfor determining the position of the object along a third axis,perpendicular to the first and second axes, determination of theposition of the object along the third axis comprising: generating athird signature signal representative of a spatial variation of theintensity of the first spot and of the second spot along the third axis,computing an auto-convolution signal by auto-convolution of the thirdsignature signal, determining a maximum of the auto-convolution signal,the coordinate of the maximum of the auto-convolution signal along thethird axis being considered as being twice the position of the beadalong the third axis.
 6. The optical device according to claim 1,wherein the optical device comprises a first light source arranged toemit light radiations toward the object according to a first angle and asecond light source arranged to emit light radiations toward the objectaccording to a second angle.
 7. The optical device according to claim 6,wherein the first light source and the first pair of apertures arearranged such that a part of the light radiations emitted by the firstlight source and diffused by the object passes through the first pair ofapertures while a part of the light radiations emitted by the firstlight source but which is not diffused by the object is blocked by thetransmission mask.
 8. The optical device according to claim 6, whereinthe second light source and the second pair of apertures are arrangedsuch that a part of the light radiations which is emitted by the secondlight source and diffused by the object passes through the second pairof apertures while a part of the light radiations emitted by the secondlight source and which is not diffused by the object is blocked by thetransmission mask.
 9. The optical device according to claim 1, whereinthe light source is a light source with a short length of coherence,such as a light emitting diode (LED).
 10. The optical device accordingto claim 1, comprising a separating arrangement for separating the firstbeams from the second beams in opposite directions along a third axis,perpendicular to the first and second axes.
 11. The optical deviceaccording to claim 10, wherein the separating arrangement comprises atleast one blade having a face which is inclined relative to a planeperpendicular to the first axis, so that the first beams or the secondbeams which go through the blade are translated along the third axis.12. The optical device according to claim 10, wherein the separatingarrangement comprises at least one prism having a face which is inclinedrelative to a plane perpendicular to the first axis, so that the firstbeams or the second beams which go through the prism are deviated alongthe third axis.
 13. The optical device according to claim 1, wherein thetransmission mask is located in a Fourier plane of the objective or in aplane which is an image of the Fourier plane of the objective through anoptical relay, so as to select parts of the light radiations which havebeen diffused by the object according to predetermined angles.
 14. Amethod for measuring a position along a first axis of an objectsubjected to light radiations emitted buy a light source, wherein themethod comprises steps of: collecting light radiations diffused by theobject by means of an objective of an optical imaging system, theobjective having an optical axis extending parallel to the first axis,arranging a transmission mask having at least a first pair of aperturesand a second pair of apertures, so as to divide the radiation diffusedby the object into two first beams passing through the first pair ofapertures and two second beams passing through the second pair ofapertures, while blocking a part of the radiations emitted by the lightsource which is not diffused by the object, the first pair of aperturesand second pair of apertures being spaced from each other along a secondaxis, perpendicular to the first axis, generating, by means of adetector having a detector plane, an image including a first spot and asecond spot, the first spot and the second spot being representative ofthe first beams and second beams impacting the detector planerespectively, the two first beams interfering with each other so as tocreate a first interference pattern within the first spot and the twosecond beams interfering with each other so as to create a secondinterference pattern within the second spot, wherein variation of theposition of the object relative to an object plane of the imaging systemalong the first axis causes spatial phase shifting of first interferencepatterns and of the second interference pattern relative to each other.15. The method according to claim 14, comprising a step of: separatingthe first beams from the second beams in opposite directions along athird axis perpendicular to the first and second axes.
 16. The methodaccording to claim 14, wherein the object is a magnetic bead.
 17. Themethod according to claim 16, wherein a molecule having two ends isattached at one end to an anchoring surface and at the other end to themagnetic bead, the device being positioned relative to the anchoringsurface so as to measure a distance between the magnetic bead and theanchoring surface.
 18. The method according to claim 17, wherein aplurality of molecules are attached, each molecule being attached at oneend to the anchoring surface and at the other end to an associatedmagnetic bead, and the method comprises a step of generating an imageshowing a plurality of pairs of spots, each pair of spots beinggenerated by one of the magnetic beads, and scanning the image fordetermining for each successive pair of spots, a distance between themagnetic bead and the anchoring surface.